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Tikhonov regularization is a widely used technique in solving inverseproblems that can enforce prior properties on the desired solution. In thispaper, we propose a Krylov subspace based iterative method for solving linearinverse problems with general-form Tikhonov regularization term $x^TMx$, where$M$ is a positive semi-definite matrix. An iterative process called thepreconditioned Golub-Kahan bidiagonalization (pGKB) is designed, whichimplicitly utilizes a proper preconditioner to generate a series of solutionsubspaces with desirable properties encoded by the regularizer $x^TMx$. Basedon the pGKB process, we propose an iterative regularization algorithm viaprojecting the original problem onto small dimensional solution subspaces. Weanalyze regularization effect of this algorithm, including the incorporation ofprior properties of the desired solution into the solution subspace and thesemi-convergence behavior of regularized solution. To overcome instabilitiescaused by semi-convergence, we further propose two pGKB based hybridregularization algorithms. All the proposed algorithms are tested on bothsmall-scale and large-scale linear inverse problems. Numerical resultsdemonstrate that these iterative algorithms exhibit excellent performance,outperforming other state-of-the-art algorithms in some cases.

A preconditioned Krylov subspace method for linear inverse problems with general-form Tikhonov regularization